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Checking for Fit
©2005 Dr. B. C. Paul
The Assumption of Normalcy
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We have learned statistical tests and
confidence intervals that work for normally
distributed data
We did ANOVA but assumed that all the
random errors between classes were equal
and normally distributed
We did Regression and assumed that all the
random error around the regression line was
normally distributed.
Do We Really Believe the Whole
World is Normally Distributed?
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Probably not
We talked about levels of effort
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1- Just have faith and assume (which we have
done a lot of lately)
2- Graphical Inspection (with our regression we
started looking at plots of residuals)
3- Actual tests and calibration
Testing to See if Data is
Normally Distributed
Must of course begin with a
Data set.
Click on Analyze to Pull Down
The menu
Highlight Descriptive Statistics
To bring out side menu
Highlight and Click Explore
Load Up the Variable to Study
In this case I brought MPG
Over to be analyzed.
Note that the default is to give me both plots and statistics
Click on the Statistics Tab to Take
a Closer Look at What We Get
I just checked off the works
The default is the descriptive
Statistics and a confidence
Interval on the mean.
Reading Output
Mean Value
95% Confidence Interval on
Mean
Variance
This Might Not Be Normal
Skewness – derived from
The sum of deviations of
Samples from the mean
Cubed. (Thus it has a
Sign – a long upper tail
Pulls positive – a long
Lower tail pulls negative)
Our Skewness is negative
(long lower tail?)
Standard error is large
Enough we cannot be
Sure this is not a sample
Quirk.
A Kase of Kurtosis
Kurtosis is a function of the
Fourth power of the deviations
Of individual scores from the
Mean.
Positive values indicate extra
Weight in the tails – it indicates
That the confidence intervals
On the standard deviations are
Wider than expected
Kurtosis 1 means about 1.5
Times normal standard deviation
Kurtosis 2 means about twice
Negative Kurtosis indicates a flat top distribution – Ie the pile at the center is too small
This distribution has a negative Kurtosis that could be somewhat flattened.
A Shifted Mean
We have Skewness and a Median
Value suggesting a long lower
Tail.
Mathematical Averages for the mean
Assume high and low values are
Equally likely. A negative tail tends
To shift likely estimates of the mean
Down.
There are several suggested ways
Of trying to compensate. (Using
them with confidence
intervals is another
matter.
Normality Testing
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Kolmogorov-Smirnov Test
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Count what percentage of the samples are within certain
distance of the estimated mean
Compare the percentage with what you would expect from
a normal distribution (or other expected distribution)
If the deviations are large the test statistic will be large
Does tend to need larger samples
Shapiro-Wilk Test
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Similar type of test but tends to be geared more for
samples under about 50 in size.
Check Our Stats
The departure from Normal is interesting
But does not meet levels normally needed
To reject a normal distribution.
Shapiro-Wilk found deviations enough to
Be suspicious
Pretty Pictures
Big Chunk in the
Lower tail may
Be our problem.
We expect the mean about here.
The Normal Probability Plot
Upper part does not look to bad, but there
Is that lower tail lump.
Spread Around the Median
Can see a negative tail (although we
Suspect a lump in the tail).
So What Does This Mean?
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The distribution is showing some suspicious
deviations from normal
Our ability to categorically reject a normal
distribution is limited (K-S test did not reject)
Problem appears to center on a clump of lower tail
values
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From a science standpoint is there a reason to believe
there may be something bimodal about this population?
 If there is no science then one may want to allow normal
test procedures to go forward (with some caution about the
lower tail if you think there might be some science to it)
Robustness
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Many of the statistics procedures are fairly robust
and yield answers fairly close to right for normal-like
populations
We are fitting models to populations – sometimes
we may fool ourselves if we assume we can get 6
place accuracy from a mathematical model with a
convenient similarity to our data
Note that when one rejects in boarder-line cases
calls for scientific judgment about what it is you are
modeling (computer can’t study your subject for
you).